IJPAM: Volume 70, No. 5 (2011)

OPTIMAL CONVEX COMBINATION BOUNDS OF
THE CENTROIDAL AND HARMONIC MEANS
FOR THE SEIFFERT MEAN

Gao Shaoqin$^1$, Gao Hongya$^2$, Shi Wenying$^3$
College of Mathematics and Computer Science
Hebei University
Baoding, 071002, P.R. CHINA


Abstract. We find the greatest value $\alpha$ and the least value $\beta$ such that the double inequality

\begin{displaymath} \alpha T(a,b)+(1-\alpha)H(a,b)<P(a,b)<\beta T(a,b)+(1-\beta)H(a,b) \end{displaymath}

holds for all $a,b>0$ with $a\neq b$. Here $T(a,b)$, $H(a,b)$ and $P(a,b)$ denote the Centroidal, harmonic, and the Seiffert means of two positive numbers $a$ and $b$, respectively.

Received: February 9, 2011

AMS Subject Classification: 26D15

Key Words and Phrases: optimal convex combination bound, Centroidal mean, harmonic mean, the Seiffert mean

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 70
Issue: 5